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Schrödinger equation

2007 Schools Wikipedia Selection. Related subjects: General Physics

   In physics, the Schrödinger equation, proposed by the Austrian
   physicist Erwin Schrödinger in 1925, describes the space- and
   time-dependence of quantum mechanical systems. It is of central
   importance to the theory of quantum mechanics, playing a role analogous
   to Newton's second law in classical mechanics.

   In the mathematical formulation of quantum mechanics, each system is
   associated with a complex Hilbert space such that each instantaneous
   state of the system is described by a unit vector in that space. This
   state vector encodes the probabilities for the outcomes of all possible
   measurements applied to the system. As the state of a system generally
   changes over time, the state vector is a function of time. The
   Schrödinger equation provides a quantitative description of the rate of
   change of the state vector.

   Using Dirac's bra-ket notation, the definition of energy results in the
   time derivative operator: at time t by
   \left|\psi\left(t\right)\right\rangle . The Schrödinger equation is

          H(t)\left|\psi\left(t\right)\right\rangle = \mathrm{i}\hbar
          \frac{\partial}{\partial t} \left| \psi \left(t\right)
          \right\rangle

   where i is the imaginary unit, t is time, \partial / \partial t is the
   partial derivative with respect to t, \hbar is the reduced Planck's
   constant (Planck's constant divided by 2π), ψ(t) is the wave function,
   and H\!\left(t\right) is the Hamiltonian (a self-adjoint operator
   acting on the state space).

   The Hamiltonian describes the total energy of the system. As with the
   force occurring in Newton's second law, its exact form is not provided
   by the Schrödinger equation, and must be independently determined based
   on the physical properties of the system.

Time-independent Schrödinger equation

   For many real-world problems the energy operator H does not depend on
   time. Then it can be shown that the time-dependent Schrödinger equation
   simplifies to the time-independent Schrödinger equation, which—as we
   will discuss—has the well-known appearance HΨ = EΨ.

   An example of a simple one-dimensional time-independent Schrödinger
   equation for a particle of mass m, moving in a potential U(x) is:

                -\frac{\hbar^2}{2 m} \frac{d^2 \psi (x)}{dx^2} + U(x) \psi
                (x) = E \psi (x).

   The analogous 3-dimensional time-independent equation is, :

                \left[-\frac{\hbar^2}{2 m} \nabla^2 + U(r) \right] \psi
                (r) = E \psi (r).

   For every time-independent Hamiltonian, H, there exists a set of
   quantum states, \left|\psi_n\right\rang , known as energy eigenstates,
   and corresponding real numbers E[n] satisfying the eigenvalue equation,

                H \left|\psi_n\right\rang = E_n \left|\psi_n \right\rang.

   Such a state possesses a definite total energy, whose value E[n] is the
   eigenvalue of the Hamiltonian. The corresponding eigenvector \psi_n\,
   is normalizable to unity. This eigenvalue equation is referred to as
   the time-independent Schrödinger equation. We purposely left out the
   variable(s) on which the wavefunction \psi_n\, depends. In the first
   example above it depends on the single variable x and in the second on
   x, y, and z—the components of the vector r. In both cases the
   Schrödinger equation has the same appearance, but its Hamilton operator
   is defined on different function (state, Hilbert) spaces. In the first
   example the function space consists of functions of one variable and in
   the second example the function space consists of functions of three
   variables.

   Self-adjoint operators, such as the Hamiltonian, have the property that
   their eigenvalues are always real numbers, as we would expect, since
   the energy is a physically observable quantity. Sometimes more than one
   linearly independent state vector corresponds to the same energy E[n].
   If the maximum number of linearly independent eigenvectors
   corresponding to E[n] equals k, we say that the energy level E[n] is
   k-fold degenerate. When k=1 the energy level is called non-degenerate.

   On inserting a solution of the time-independent Schrödinger equation
   into the full Schrödinger equation, we get

          \mathrm{i} \hbar \frac{\partial}{\partial t} \left| \psi_n
          \left(t\right) \right\rangle = E_n
          \left|\psi_n\left(t\right)\right\rang.

   It is relatively easy to solve this equation. One finds that the energy
   eigenstates (i.e., solutions of the time-independent Schrödinger
   equation) change as a function of time only trivially, namely, only by
   a complex phase:

          \left| \psi \left(t\right) \right\rangle =
          \mathrm{e}^{-\mathrm{i} Et / \hbar}
          \left|\psi\left(0\right)\right\rang.

   It immediately follows that the probability amplitude,

                \psi(t)^*\psi(t) = \mathrm{e}^{\mathrm{i} Et /
                \hbar}\mathrm{e}^{-\mathrm{i} Et / \hbar} \psi(0)^*\psi(0)
                = |\psi(0)|^2,

   is time-independent. Because of a similar cancellation of phase factors
   in bra and ket, all average (expectation) values of time-independent
   observables (physical quantities) computed from \psi(t)\, are
   time-independent.

   Energy eigenstates are convenient to work with because they form a
   complete set of states. That is, the eigenvectors
   \left\{\left|n\right\rang\right\} form a basis for the state space. We
   introduced here the short-hand notation |\,n\,\rang = \psi_n . Then any
   state vector that is a solution of the time-dependent Schrödinger
   equation (with a time-independent H)
   \left|\psi\left(t\right)\right\rang can be written as a linear
   superposition of energy eigenstates:

          \left|\psi\left(t\right)\right\rang = \sum_n c_n(t)
          \left|n\right\rang \quad,\quad H \left|n\right\rang = E_n
          \left|n\right\rang \quad,\quad \sum_n
          \left|c_n\left(t\right)\right|^2 = 1.

   (The last equation enforces the requirement that
   \left|\psi\left(t\right)\right\rang , like all state vectors, may be
   normalized to a unit vector.) Applying the Hamiltonian operator to each
   side of the first equation, the time-dependent Schrödinger equation in
   the left-hand side and using the fact that the energy basis vectors are
   by definition linearly independent, we readily obtain

          \mathrm{i}\hbar \frac{\partial c_n}{\partial t} = E_n
          c_n\left(t\right).

   Therefore, if we know the decomposition of
   \left|\psi\left(t\right)\right\rang into the energy basis at time t =
   0, its value at any subsequent time is given simply by

          \left|\psi\left(t\right)\right\rang = \sum_n
          \mathrm{e}^{-\mathrm{i}E_nt/\hbar} c_n\left(0\right)
          \left|n\right\rang.

   Note that when some values c_n(0)\, are not equal to zero for differing
   energy values E_n\, , the left-hand side is not an eigenvector of the
   energy operator H. The left-hand is an eigenvector when the only
   c_n(0)\, -values not equal to zero belong the same energy, so that
   \mathrm{e}^{-\mathrm{i}E_nt/\hbar} can be factored out. In many
   real-world application this is the case and the state vector \psi(t)\,
   (containing time only in its phase factor) is then a solution of the
   time-independent Schrödinger equation.

Example

   Let |\,1\,\rangle and |\,2\,\rangle be degenerate eigenstates of the
   time-independent Hamiltonian H\, :

          H\,|\,1\,\rangle = E |\,1\,\rangle \quad \hbox{and} \quad
          H\,|\,2\,\rangle = E |\,2\,\rangle,

   Suppose at t = 0 a solution \Psi(t)\, of the full Schrödinger equation
   has the form

          |\,\Psi(t)\,\rangle = \mathrm{e}^{-\mathrm{i}E/\hbar}
          c_1\left(0\right) |\,1\,\rangle +
          \mathrm{e}^{-\mathrm{i}E/\hbar} c_2\left(0\right) |\,2\,\rangle,

   then

          H\,|\,\Psi(t)\,\rangle = E \mathrm{e}^{-\mathrm{i}E/\hbar}
          c_1\left(0\right) |\,1\,\rangle +
          E\mathrm{e}^{-\mathrm{i}E/\hbar} c_2\left(0\right) |\,2\,\rangle
          = E\left(\mathrm{e}^{-\mathrm{i}E/\hbar} c_1\left(0\right)
          |\,1\,\rangle + \mathrm{e}^{-\mathrm{i}E/\hbar}
          c_2\left(0\right) |\,2\,\rangle\right)

                            = E\,|\,\Psi(t)\,\rangle.

   Conclusion: The wavefunction \Psi(t)\, with the given initial condition
   (its form at t = 0), remains a solution of the time-independent
   Schrödinger equation for all times t.

Footnote

    1. ^ In fact also an initial condition must be used here. At time zero
       the wavefunction must be an eigenstate of H.

Schrödinger wave equation

   The state space of certain quantum systems can be spanned with a
   position basis. In this situation, the Schrödinger equation may be
   conveniently reformulated as a partial differential equation for a
   wavefunction, a complex scalar field that depends on position as well
   as time. This form of the Schrödinger equation is referred to as the
   Schrödinger wave equation.

   Elements of the position basis are called position eigenstates. We will
   consider only a single-particle system, for which each position
   eigenstate may be denoted by \left|\mathbf{r}\right\rang , where the
   label \mathbf{r} is a real vector. This is to be interpreted as a state
   in which the particle is localized at position \mathbf{r} . In this
   case, the state space is the space of all square-integrable complex
   functions.

The wavefunction

   We define the wavefunction as the projection of the state vector
   \left|\psi\left(t\right)\right\rang onto the position basis:

          \psi\left(\mathbf{r}, t\right) \equiv \left\langle \mathbf{r} |
          \psi\left(t\right) \right\rangle.

   Since the position eigenstates form a basis for the state space, the
   integral over all projection operators is the identity operator:

          \int \left|\mathbf{r}\right\rangle \left\langle \mathbf{r}
          \right| \mathrm{d}^3 \mathbf{r} = \mathbf{I}.

   This statement is called the resolution of the identity. With this, and
   the fact that kets have unit norm, we can show that
   \left\lang \psi(t) | \psi(t) \right\rang = \left\lang
   \psi\left(t\right) \right| \; \left(\int \; \left|\mathbf{r}\right\rang
   \lang\mathbf{r}| \; \mathrm{d}^3\mathbf{r} \right)
   \left|\psi\left(t\right)\right\rang
   = \int \; \left\lang\psi\left(t\right)|\mathbf{r}\right\rang
   \left\lang\mathbf{r}|\psi\left(t\right) \right\rang \; \mathrm{d}^3
   \mathbf{r}
   = \int \; \psi\left(\mathbf{r}, t\right)^* \; \psi\left(\mathbf{r},
   t\right) \; \mathrm{d}^3\mathbf{r}
   = 1\;

   where \psi\left(\mathbf{r}, t\right)^* denotes the complex conjugate of
   \psi\left(\mathbf{r}, t\right) . This important result tells us that
   the absolute square of the wavefunction, integrated over all space,
   must be equal to 1:

          \int \; \left|\psi\left(\mathbf{r}, t\right)\right|^2 \;
          \mathrm{d}^3\mathbf{r} = 1.

   We can thus interpret the absolute square of the wavefunction as the
   probability density for the particle to be found at each point in
   space. In other words, \left|\psi\left(\mathbf{r}, t\right)\right|^2
   \mathrm{d}^3\mathbf{r} is the probability, at time t, of finding the
   particle in the infinitesimal region of volume \mathrm{d}^3\mathbf{r}
   surrounding the position \mathbf{r} .

   We have previously shown that energy eigenstates vary only by a complex
   phase as time progresses. Therefore, the absolute square of their
   wavefunctions do not change with time. Energy eigenstates thus
   correspond to static probability distributions.

Operators in the position basis

   Any operator A acting on the wavefunction is defined in the position
   basis by

          A \psi\left(\mathbf{r}, t\right) \equiv \left\lang\mathbf{r}| A
          | \psi\left(t\right) \right\rang.

   The operators A on the two sides of the equation are different things:
   the one on the right acts on kets, whereas the one on the left acts on
   scalar fields. It is common to use the same symbols to denote operators
   acting on kets and their projections onto a basis. Usually, the kind of
   operator to which one is referring is apparent from the context, but
   this is a possible source of confusion.

   Using the position-basis notation, the Schrödinger equation can be
   written as

          H \psi\left(\mathbf{r},t\right) = \mathrm{i} \hbar
          \frac{\partial}{\partial t} \psi\left(\mathbf{r},t\right).

   This form of the Schrödinger equation is the Schrödinger wave equation.
   It may appear that this is an ordinary differential equation, but in
   fact the Hamiltonian operator typically includes partial derivatives
   with respect to the position variable \mathbf{r} . This usually leaves
   us with a difficult linear partial differential equation to solve.

Non-relativistic Schrödinger wave equation

   In non-relativistic quantum mechanics, the Hamiltonian of a particle
   can be expressed as the sum of two operators, one corresponding to
   kinetic energy and the other to potential energy. The Hamiltonian of a
   particle with no electric charge and no spin in this case is:

          H \psi\left(\mathbf{r}, t\right) = \left(T + V\right) \,
          \psi\left(\mathbf{r}, t\right) = \left[ - \frac{\hbar^2}{2m}
          \nabla^2 + V\left(\mathbf{r}\right) \right]
          \psi\left(\mathbf{r}, t\right) = \mathrm{i} \hbar \frac{\partial
          \psi}{\partial t} \left(\mathbf{r}, t\right)

                where

                      T = \frac{p^2}{2m} is the kinetic energy operator,
                      m is the mass of the particle,
                      \mathbf{p} = -\mathrm{i}\hbar\nabla is the momentum
                      operator,
                      V = V\left(\mathbf{r}\right) is the potential energy
                      operator,
                      V is a real scalar function of the position operator
                      \mathbf{r} ,
                      \nabla is the gradient operator, and
                      \nabla^2 is the Laplace operator.

   This is a commonly encountered form of the Schrödinger wave equation,
   though not the most general one. The corresponding time-independent
   equation is

          \left[ - \frac{\hbar^2}{2m} \nabla^2 + V\left(\mathbf{r}\right)
          \right] \psi\left(\mathbf{r}\right) = E \psi
          \left(\mathbf{r}\right).

   The relativistic generalisations of this wave equation are the Dirac
   equation, Klein-Gordon equation, Proca equation, Maxwell equations etc,
   depending on spin and mass of the particle. See relativistic wave
   equations for details.

Probability currents

   In order to describe how probability density changes with time, it is
   acceptable to define probability current or probability flux. The
   probability flux represents a flowing of probability across space.

   For example, consider a Gaussian probability curve centered around x[0]
   with x[0] moving at speed v to the right. One may say that the
   probability is flowing toward right, i.e., there is a probability flux
   directed to the right.

   The probability flux \mathbf{j} is defined as:

                \mathbf{j} = {\hbar \over m} \cdot {1 \over {2
                \mathrm{i}}} \left( \psi ^{*} \nabla \psi - \psi \nabla
                \psi^{*} \right) = {\hbar \over m} \operatorname{Im}
                \left( \psi ^{*} \nabla \psi \right)

   and measured in units of (probability)/(area × time) = r^−2t^−1.

   The probability flux satisfies a quantum continuity equation, i.e.:

                { \partial \over \partial t} P\left(x,t\right) + \nabla
                \cdot \mathbf{j} = 0

   where P\left(x, t\right) is the probability density and measured in
   units of (probability)/(volume) = r^−3. This equation is the
   mathematical equivalent of probability conservation law.

   It is easy to show that for a plane wave,

          \left| \psi \right\rang = A e^{ \mathrm{i} k x} e^{ - \mathrm{i}
          \omega t}

   the probability flux is given by

          j\left(x,t\right) = \left|A\right|^2 {k \hbar \over m}.

Solutions of the Schrödinger equation

   Analytical solutions of the time-independent Schrödinger equation can
   be obtained for a variety of relatively simple conditions. These
   solutions provide insight into the nature of quantum phenomena and
   sometimes provide a reasonable approximation of the behaviour of more
   complex systems (e.g., in statistical mechanics, molecular vibrations
   are often approximated as harmonic oscillators). Several of the more
   common analytical solutions include:
     * The free particle
     * The particle in a box
     * The finite potential well
     * The Delta function potential
     * The particle in a ring
     * The particle in a spherically symmetric potential
     * The quantum harmonic oscillator
     * The linear rigid rotor
     * The symmetric top
     * The hydrogen atom or hydrogen-like atom
     * The ring wave guide
     * The particle in a one-dimensional lattice (periodic potential)

   For many systems, however, there is no analytic solution to the
   Schrödinger equation. In these cases, one must resort to approximate
   solutions. Some of the common techniques are:
     * Perturbation theory
     * The variational principle underpins many approximate methods (like
       the popular Hartree-Fock method which is the basis of the post
       Hartree-Fock methods)
     * Quantum Monte Carlo methods
     * Density functional theory
     * The WKB approximation
     * discrete delta-potential method

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